Similar as that in Ref. [1]}, the diagonal term \(\frac{\partial \Omega ^2}{\partial {\bf k}}\)  in the bracket
corresponding to \(\omega _\sigma \frac{\partial \omega _\sigma }{ \partial {\bf k}}\)  has no contribution to the phonon Hall conductivity because which
is an odd function of \({\bf k}\) . Because of the off-diagonal term
\([\mathcal {A}_{k_\alpha },\Omega ^2]_{\sigma ,\sigma ^{\prime }}=(\omega _{\sigma ^{\prime }}^2-\omega _{\sigma }^2)\mathcal {A}_{k_\alpha }^{\sigma \sigma ^{\prime }}\) 
